An argument analogous to the one in section 2 shows that \(\Pi (a,n)\)  does
not vanish for \(|\chi _{3}\rangle \)  or for \(|\chi _{4}\rangle \) . In light of
Eq. (REF ), for \(n>2\) , the states \(|\chi _{3}\rangle \)  and \(|\chi _{4}\rangle \)  are both different from the \(|GHZ\rangle \) , \(|W\rangle \) , and
Dicke states under SLOCC. It can be further demonstrated that \(|\chi _{3}\rangle \)  and \(|\chi _{4}\rangle \)  are entangled, and that \(|\chi _{4}\rangle \)  is equivalent to \(|\chi _{3}\rangle \)  under SLOCC. We
exemplify the result for the case of four qubits. We find that \(|\chi _{3}\rangle =(1/2)\bigl (|0\rangle +|3\rangle +|12\rangle -|15\rangle \bigr )\) 
and it was shown in [1]} that \(|\chi _{3}\rangle \)  is different
from the \(|GHZ\rangle \) , \(|W\rangle \) , and Dicke states under SLOCC.
